# Investigate Each Limit...(A)

nycmathguy
Homework Statement:
Investigate each limit.
Relevant Equations:
See attachment for function.
Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

#### Attachments

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Mark44
Mentor
Homework Statement:: Investigate each limit.
Relevant Equations:: See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.
Plot some points. What are f(0), f(1/4), f(1/2), f(2), f(2.5), etc.?

Also, you've posted a few threads just now with little or no work shown. That's a violation of forum rules. You have to show some effort. You have the formula for the function -- sketch a graph of it.

Delta2
Homework Helper
Gold Member
Focus at an interval [n,n+1] where n is an integer. Answer the following questions to help you understand how this function goes
1) What is f(n)
2) What is f(n+1)
3) What is f(x) for every ##x\in(n,n+1)## for example for x=(2n+1)/2 the midpoint of n and n+1.

nycmathguy
Focus at an interval [n,n+1] where n is an integer. Answer the following questions to help you understand how this function goes
1) What is f(n)
2) What is f(n+1)
3) What is f(x) for every ##x\in(n,n+1)## for example for x=(2n+1)/2 the midpoint of n and n+1.
Sorry but I don't get it. Still lost.

Delta2
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Gold Member
what is f(1) and f(2) equal to for example? Hint: 1 and 2 are integers

nycmathguy
Delta2
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Gold Member
The definition of the function f(x) tells you that f(x)=1 if x is integer and f(x)=0 if x is not integer.

nycmathguy
nycmathguy
The definition of the function f(x) tells you that f(x)=1 if x is integer and f(x)=0 if x is not integer.

I say for (1), the answer is 0.
The answer for (2) is 1.

Yes?

Delta2
Homework Helper
Gold Member
I say for (1), the answer is 0.
The answer for (2) is 1.

Yes
Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.

nycmathguy
Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.
For (1), x tends to an integer. Thus, then f(x) = 1.

For (2), x tends to a rational number. Thus, f(x) = 0.

Delta2
Homework Helper
Gold Member
Yes but as x tends to an integer, it passes from all sorts of rationals and irrationals (from the left and right of integer) for which f(x)=0.

nycmathguy
Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.
What about the following two cases using the same attachment?

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.

Yes?

Delta2
Homework Helper
Gold Member
If I give you the following definition for f:
f(1)=f(0)=1
f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the ##\lim_{x\to 0} f(x)## (or ##\lim_{x\to 1} f(x)##..

nycmathguy
Delta2
Homework Helper
Gold Member
What about the following two cases using the same attachment?

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.

Yes?
For both cases the limit is 0. (0 is an integer btw).

nycmathguy
Homework Statement:: Investigate each limit.
Relevant Equations:: See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

If I give you the following definition for f:
f(1)=f(0)=1
f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the ##\lim_{x\to 0} f(x)## (or ##\lim_{x\to 1} f(x)##..
Can you elaborate a little more?
It's just not sinking in. In fact, Sullivan stated in his book that this is considered a challenging problem.

Delta2
Homework Helper
Gold Member
hm ok let me see
If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.

nycmathguy
Delta2
Homework Helper
Gold Member
I think you are confusing the ##\lim_{x\to x_0}f(x)## with the ##f(x_0)##. These two are equal only if the function f is continuous at ##x_0##. But in this problem here we have to deal with a function f that is not continuous at every integer.

nycmathguy
nycmathguy
hm ok let me see
If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.
In that case, it is 0.

Delta2
Delta2
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Gold Member
In that case, it is 0.
Correct now lets say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?

nycmathguy
Correct now lets say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?
You said except for x = 0. I say the limit is 1?

Delta2
Homework Helper
Gold Member
You said except for x = 0. I say the limit is 1?
Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..

nycmathguy
nycmathguy
Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..
So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.

Delta2
Delta2
Homework Helper
Gold Member
So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.
That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?

nycmathguy
That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?
So, f(x) tends to 0 as x-->0.

Delta2
Delta2
Homework Helper
Gold Member
So, f(x) tends to 0 as x-->0.
yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all ##x\neq 0## .

nycmathguy
nycmathguy
yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all ##x\neq 0## .
Trust me, I plan to journey through calculus l,ll, and lll. We will see limit questions up the wall.